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The extensive properties are additive. An intensive property is a ratio of two extensive properties. So I think we can add intensive properties.

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Its because intensive properties like temperature, refractive index, density are same for all the portions of the container of gas. You can't say that if the temperature of one portion is x and and other portion is also x, then the total temperature is $2x$.

But extensive properties, they are different for different portions of the container like mass, volume, length. So they can be added.

It doesn't matter if the intensive property is a ratio of two extensive properties. If two extensive properties are in ratio then if first extensive property changes then the second one should change.

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An intensive property is a ratio of two extensive properties. So I think we can add intensive properties.

This is a tempting thought but is wrong for several reasons.

Mathematical inconsistency

This doesn't make mathematical sense. Consider two extensive properties for system 1, called $A$ and $B$, and let intensive property $X = \frac{A}{B}$. Consider another system, system 2, for which the two extensive properties are $a$ and $b$. The corresponding intensive property is $x=\frac{a}{b}$.

The question is, what are the properties of the combined systems 1 and 2 when considered as one single larger system? You seem to be saying that you can approximate $X + x \approx \frac{A + a}{B + b}$ but this approximation is wrong.

Instead, $X + x = \frac{A}{B} + \frac{a}{b} = \frac{bA + Ba}{Bb}$. So adding the intensive properties is not the same as adding each extensive property separately and then taking the ratio again.

Conceptual inconsistencies

It is true that all ratios of two extensive properties are intensive properties. But I don't think it's true that all intensive properties are easily defined as the ratio of two extensive properties. For example, temperature is an intensive property, but it isn't at all obvious to me what two extensive properties temperature is a "ratio" of.

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  • $\begingroup$ I remember Richard Feynman's book Surely you are joking, Mr. Feynman!, where he quotes from some Brasilian textbook of physics brought to him to review it. It was listing surface temperatures of stars of various colours, asking what is their total temperature. :-D $\endgroup$
    – Poutnik
    Commented Jan 20, 2023 at 19:14
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    $\begingroup$ $E_\mathrm{therm}=\frac 32 N k_\mathrm{B} T \implies T = \frac{2}{3k_\mathrm{B}}\left(\frac{E_\mathrm{therm}}{N}\right)$, where $N$ is the number of degrees of freedom. Both quantities are extensive. $\endgroup$
    – Poutnik
    Commented Jan 20, 2023 at 20:17

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